On the realization of exact three-point difference schemes for second- order ordinary differential equations with piecewise smooth coefficients. (English. Russian original) Zbl 0717.65052

Sov. Math., Dokl. 41, No. 3, 463-467 (1990); translation from Dokl. Akad. Nauk SSSR 312, No. 3, 538-543 (1990).
On the basis of an exact three-point difference scheme for a boundary value problem of the form \(L^{(k,q)}u=-f(x),\quad x\in (0,1),\quad u(0)=A,\quad u(1)=B,\quad 0<C_ 1\leq k(x)\leq C_ 2,\quad | q(x)| \leq C_ 2\) in the class of piecewise smooth functions k(x), q(x) and f(x), the authors construct a three-point difference scheme of an arbitrary order of accuracy. To define this scheme at an arbitrary point \(x_ j\) of the mesh \(w_ h\) it is necessary to solve four auxiliary Cauchy problems: two on the interval \([x_{j-1},x_ j]\) and two on the interval \([x_ j,x_{j+1}]\). Each of the Cauchy problems is solved in one step by a one-step method.
Reviewer: A.Marciniak


65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations